The document discusses matrices and systems of linear equations. It defines a matrix as a rectangular array of numbers and explains how matrices can organize data. It also defines an augmented matrix, which contains the coefficients and constants of a system of linear equations. Elementary row operations are performed on the augmented matrix to put it in row-echelon form and determine whether the system has a unique solution, no solution, or infinitely many solutions.

1. VCLA
SYSTEM OF LINEAR EQUATION AND
MATRICES
Adani Institute of Infrastructure
Engineering
L-3 Group:
Rajvir Solanki - 161310109046
Raksha Agarwal - 161310109047
Nirav Rami - 161310109048
Arpit Raval - 161310109050
Rohan Kaushik - 161310109051

2. MATRICES
 A matrix is simply a rectangular array of numbers.
 Matrices are used to organize information into
categories that correspond to the rows and columns
of the matrix.
 This is a compact way of saying there are 12
immature males, 15 immature females, 18 adult
males, and so on.

3. LINEAR SYSTEM OF MATRICES
 This matrix is called the augmented matrix of the
system.
 The augmented matrix contains the same
information as the system, but in a simpler form.
 The operations we learned for solving systems of
equations can now be performed on the
augmented matrix.

4. AUGMENTED MATRIX
 We can write a system of linear
equations as a matrix by writing only
the coefficients and constants that appear in the
equations.
 This is called the augmented matrix
of the system.
Linear System Augmented Matrix
3 2 5
3 0
4 11
x y z
x y z
x z
  

  
  
3 2 1 5
1 3 1 0
1 0 4 11
 
  
  

5. Elementary Row Operations
1. Add a multiple of one row to another.
2. Multiply a row by a nonzero constant.
3. Interchange two rows.
 Note that performing any of these
operations on the augmented matrix of
a system does not change its solution.

6. ROW-ECHELON FORM
A matrix is in row-echelon form if it
satisfies the following conditions.
1. The first nonzero number in each row
(reading from left to right) is 1.
This is called the leading entry.
2. The leading entry in each row is to the right of
the leading entry in the row immediately above it.
3. All rows consisting entirely of zeros are at
the bottom of the matrix.

7. EXAMPLE
12215
15432
4111








12215
7210
4111



32760
7210
4111



32760
7210
4111



10500
7210
4111



2100
7210
4111




8. CONSISTENT OR INCONSISTENT?
2100
5010
1001
2000
7010
4001
0000
2310
2501

9. EXAMPLE FOR NO SOLUTION
4331
6212
3121








4331
0450
3121



1450
0450
3121



1450
010
3121
5
4



1000
010
3121
5
4



10. EXAMPLE FOR INFINITELY MANY SOLUTIONS
2 3 4 10
3 3 4 15
2 2 6 8 10
x y z w
x y z w
x y z w
   

   
    
2 1 2
3 1 3
3 2 3 1 2 1
R R R
R 2R R
R 2R R R 2R R
1 2 3 4 10 1 2 3 4 10
1 3 3 4 15 0 1 0 0 5
2 3 6 8 10 0 2 0 0 10
1 2 3 4 10 1 0 3 4 0
0 1 0 0 5 0 1 0 0 5
0 0 0 0 0 0 0 0 0 0
 
 
   
      
       
        



      
   
   

 


    
1 2 3 4 10
1 3 3 4 15
2 3 6 8 10
  
   
   
1 2 3 4 10
0 1 0 0 5
0 2 0 0 10
  
 
 
   
1 2 3 4 10
0 1 0 0 5
0 0 0 0 0
  
 
 
  
1 0 3 4 0
0 1 0 0 5
0 0 0 0 0
  
 
 
  
Thus, the complete solution is:
x = 3s + 4t
y = 5
z = s
w = t
where s and t are any real numbers.

11. KEY CONCEPTS